If there's an exceptionally talented child, who displays an excellent work ethic, can demonstrate the techniques with a high degree of proficiency, has full command of the requirements, and has proven himself to the dojo, then I wouldn't have any hesitations about awarding him a full fledged shodan rank. There simply wouldn't be anything to be gained by holding him back, if he were qualified in all of the above criteria.
However, such cases are not common at all, and if a dojo is handing out black belts to young children at too frequent of a rate, then I would suspect something is a bit amiss. Maybe I'm wrong; maybe the school is that good, and their students fit all of the above criteria, but the odds aren't exactly against my beliefs here.
Where I would draw the line, though, is when it comes to them becoming full fledged instructors. I wouldn't allow children to be in charge of teaching a class, and especially not in an unsupervised manner. Being an instructor requires significant real-life experience, and nothing can really replace that.
I wouldn't hesitate to use such children black belts, though, as excellent examples in class, and would even allow them to help out a small section of students with specific instructions. However, if one wants to become a full-fledged instructor, I'd have them wait until they've reached at least 16 years of age.